Metapopulations Objectives –Determine how e and c parameters influence metapopulation dynamics –Determine how the number of patches in a system affects the probability of local extinction and probability of regional extinction –Compare ‘propagule rain’ vs. ‘internal colonization’ metatpopulation dynamics –Evaluate how the ‘rescue effect’ affects metapopulation dynamics

Metapopulations There are relatively few examples where the entire population resides within a single patch Most species are patchily distributed across space Hence it becomes a population of populations…metapopulations

Metapopulations Metapopulation theory (Levin 1969, 1970) describes a network of patches, some occupied some not, where subpopulations are interacting (“winking”) The classic model is then based upon presence-absence, not a demographic model like source-sink dynamics

Fragmentation & Heterogeneity

Metapopulations Let’s define extinction and colonization mathematically Extinction p e thus persistence is 1-p e Colonization is p i with vacancy is 1-p i We can consider the fate of a single patch over time or the entire metapopulation over time

Metapopulations For a given patch, the likelihood of persistence for n years is simply p n = (1 – p e ) n E.g. if a patch has a probability of persistence = 0.8 in a given year, the probability for 3 years = = If we had 100 patches, approximately 52 would persist and 48 would go extinct

Metapopulations To consider the fate of the entire metapopulation (i.e. the probability of extinction of the entire population) If all patches have the same probability of extinction, it is simply p e x For example, if p e =0.5 across 6 patches then P x = 1-(p e ) x or = or 1.5%

Metapopulations Now that we have defined e and c, let us consider the basic metapopulation model where f is the fraction of patches occupied in the system (e.g. 5/25 = 0.2) If f is the fraction of patches occupied, then 1-f is the fraction empty, the we can compute I as I = p i ( 1 - f ) df / dt = I -E

Metapopulations Focusing on E, the rate at which occupied patches go extinct E should depend upon the number of patches occupied as well as the extinction probability (p e ) Substituting our new values of I and E E = p e f df/dt = p i (1 – f) - p e f

Metapopulations This is called the propagule rain model or an island-mainland model, because the colonization rate does NOT depend on patch occupancy patterns-it is assumed that colonists are available to populate and empty patch and they can come from within or outside the metapopulation df/dt = p i (1 – f) - p e f

Metapopulations At equilibrium the fraction of patches is constant, although the exact combination is dynamic The equilibrium fraction can be derived by setting the ‘rate’ to 0 dt/dt = 0 = p i – p i f –p e f and then f = p i / (p i + p e )

Metapopulations There are many important assumptions (as with any model), the most important being all patches are created equal; p e and p i are constant over time and apply to patches irrespective of population size and finally, spatial arrange and proximity are not important to p e or p i You probably can imagine how when f is high, p i is probably large

Metapopulations This type of model is called the internal colonization model because colonization rates depend on current status (f) of the metapopulation system Extinction of a patch may depend on the fraction of patches occupied in the system When f is high, there are many potential colonists and p e decreases This is termed the rescue effect

Metapopulations Rescue effect and Internal Colonization Model

Metapopulations Objectives –Determine how e and c parameters influence metapopulation dynamics –Determine how the number of patches in a system affects the probability of local extinction and probability of regional extinction –Compare ‘propagule rain’ vs. ‘internal colonization’ metatpopulation dynamics –Evaluate how the ‘rescue effect’ affects metapopulation dynamics